The aim of this review paper is to describe in a comprehensive manner the current understanding of the radiation response of state-of-the-art Silicon-on-Insulator (SOI) and FinFET CMOS technologies. Total Ionizing Dose (TID) response, heavy-ion microdose effects and single-event effects (SEEs) will be discussed. It is shown that a very high TID tolerance can be achieved by narrow-fin SOI FinFET architectures, while bulk FinFETs may exhibit similar TID response to the planar devices. Due to the vertical nature of FinFETs, a specific heavy-ion response can be obtained, whereby the angle of incidence becomes highly important with respect to the vertical sidewall gates. With respect to SEE, the buried oxide in the SOI FinFETs suppresses the diffusion tails from the charge collection in the substrate compared to the planar bulk FinFET devices. Channel lengths and fin widths are now comparable to, or smaller than the dimensions of the region affected by the single ionizing ions or lasers used in testing. This gives rise to a high degree of sensitivity to individual device parameters and source-drain shunting during ion-beam or laser-beam SEE testing. Simulations are used to illuminate the mechanisms observed in radiation testing and the progress and needs for the numerical modeling/simulation of the radiation response of advanced SOI and FinFET transistors are highlighted.